Series and Approximations in symbolic computation with SageMath

In this lecture we compute Taylor series expansions of functions at a point. An application in two variables solves the problem of computing the tangent line to a curve at a point. Viewed not as expressions, but as power series we can compute the multiplicative inverse if the constant term of the series is nonzero. Taylor series can thus approximate functions by truncation to a polynomial, but the global convergence is often not that accurate. Rational approximation have the same local convergence as power series but have a much better global, and even uniform convergence. The constructions in this lecture are illustrated in a Jupyter notebook with SageMath and with an illustration of the application of a generator made by the SymPy series.